Make the pastry: Combine the flour and salt in a large bowl and rub in the butter until the mixture resembles coarse meal. Add the beaten egg and toss with a fork, gradually adding 1 to 3 tablespoons more of cold water, until the dough can be gathered into a ball.

Halve the dough and on a board lightly dusted with flour, and pat each half into a 4- to 5-inch disk. Wrap each disk separately in plastic wrap and chill for at least 1 hour or as long as 24 hours before using.

Make the filling: Heat 1 tablespoon of the oil in a large skillet over medium heat. Add the onion, carrot, and salt and pepper to taste and cook, stirring occasionally, for 6 to 8 minutes, or until the vegetables begin to soften. Transfer the vegetables to a bowl and set aside.

Heat the remaining 2 tablespoons of oil in the same skillet over medium-high heat. Add the meat and cook, stirring often, until browned, 2 to 3 minutes. Sprinkle in the flour, reduce the heat to medium, and cook for 30 seconds, stirring constantly. Slowly add the stock, stirring to deglaze the pan. Return the vegetables to the skillet, add the mint and thyme, and season to taste with salt and pepper. Reduce the heat to medium-low, cover, and simmer until the meat is tender, about 45 minutes. Uncover the skillet and cook until thickened, about 5 minutes more. Set aside to let cool.

Assemble and bake the pies: Preheat the oven to 400.

Roll out half of the pastry on a lightly floured board into a disk about 1/4 inch thick. Cut the disk into 8 rounds about 4 inches in diameter with a cookie cutter or the floured rim of a glass. Repeat the process with the remaining pastry to make 16 rounds in all.

Put about one-eighth of the meat mixture in the middle of a pastry round, then wet the edges with a bit of water. Top with a second pastry round, then press down the edges and crimp with the tines of a fork. Cut a slit in the top of the pie and transfer to a parchment paper-lined baking sheet. Repeat the process with the remaining meat filling and pastry, arranging the pies on the baking sheet about 1 inch apart. Brush the tops of the pies with milk and bake until golden brown, about 40 minutes. Serve hot or at room temperature.